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Coutsias, et al. presented a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors. [2] They proved that the quaternion method is equivalent to the well-known Kabsch algorithm . [ 3 ]
Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent. [1] RMSD is always non-negative, and a value of 0 (almost never achieved in practice) would indicate a perfect fit to the data. In general, a lower RMSD is better than a higher one.
SNOPT is mainly written in Fortran, but interfaces to C, C++, Python and MATLAB are available. It employs a sparse sequential quadratic programming (SQP) algorithm with limited-memory quasi-Newton approximations to the Hessian of the Lagrangian. It is especially effective for nonlinear problems with functions and gradients that are expensive to ...
[1] [2] Instead of the alpha carbons typically used in structural alignment, SSAP constructs its vectors from the beta carbons for all residues except glycine, a method which thus takes into account the rotameric state of each residue as well as its location along the backbone. SSAP works by first constructing a series of inter-residue distance ...
[2] The application of secondary chemical shifts to characterize protein flexibility is based on an assumption that the proximity of chemical shifts to random coil values is a manifestation of increased protein mobility, while significant differences from random coil values are an indication of a relatively rigid structure.
Source code for algorithm implementations, and TLE interpretation in some cases: python-sgp4 A Python Implementation of the sgp4 model with automatic downloading of TLE Elements from NORAD database. PHP5 based on Gpredict; Java: SDP4 and predict4java; C++, FORTRAN, Pascal, and MATLAB. go-satellite GoLang implementation of SGP4 model and helper ...
A very high-level description of Isomap algorithm is given below. Determine the neighbors of each point. All points in some fixed radius. K nearest neighbors. Construct a neighborhood graph. Each point is connected to other if it is a K nearest neighbor. Edge length equal to Euclidean distance. Compute shortest path between two nodes. Dijkstra ...